Problem: Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle LOM = 7x - 29$, and $ m \angle MON = 7x - 35$, find $m\angle LOM$. $O$ $L$ $N$ $M$
Answer: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {7x - 29} + {7x - 35} = {90}$ Combine like terms: $ 14x - 64 = 90$ Add $64$ to both sides: $ 14x = 154$ Divide both sides by $14$ to find $x$ $ x = 11$ Substitute $11$ for $x$ in the expression that was given for $m\angle LOM$ $ m\angle LOM = 7({11}) - 29$ Simplify: $ {m\angle LOM = 77 - 29}$ So ${m\angle LOM = 48}$.